Let $x_1,\dots,x_n$ be i.i.d. drawn from $N(0,I_{p\times p})$. Consider the sample covariance matrix $W(n,p)=\frac 1n \sum_{i=1}^n x_ix_i^T$, a Wishart matrix.

For fixed $n,p$, what is the expected spectral norm of $W(n,p)$, $$\mathbb E \left[ \left\|\frac 1n \sum_{i=1}^n x_ix_i^T\right\|_2\right ] ?$$

I'm familiar with the asymptotic Bai/Silverstein result, but not sure whether that can be exploited in the finite case. If the exact expression is unwieldy, a reasonable upper bound may suffice.